Unit 08 - Grade 11-12 Physics

Circular Motion and Gravitation

Study motion in circles, centripetal acceleration, centripetal force, orbital motion, universal gravitation, satellite speed, period, and the connection between gravity and circular motion.

Lesson roadmap

What Students Should Master in This Unit

Circular motion connects kinematics, dynamics, and astronomy. Students learn that an object moving in a circle is accelerating even if its speed is constant, because its velocity direction keeps changing.

Describe circular motion

Use radius, period, frequency, speed, angular speed, and centripetal acceleration.

Analyze center-seeking force

Identify which real force provides the required centripetal force in each situation.

Connect gravity and orbits

Use Newton's law of gravitation to explain planetary, moon, and satellite motion.

Quick navigation

Jump to a Topic

Circular Motion Basics Period and Frequency Angular Motion Centripetal Acceleration Centripetal Force Sources of Center Force Vertical Circles Banked Curves Universal Gravitation Orbital Motion Simulation Labs Practice Problems
Motion in a circle

1. Circular Motion Basics

An object is in circular motion when it moves along a circular path. Even if the speed is constant, velocity changes because direction changes. A change in velocity means acceleration.

Circular motion direction diagram Circular motion has two key directions r v tangent a_c inward center
Velocity points tangent to the path, while centripetal acceleration points inward toward the center.
Circumference C = 2πr Distance traveled in one complete circle.
Tangential speed v = distance / time Speed along the circular path.
One revolution speed v = 2πr / T T is the period for one complete revolution.

Velocity and Acceleration Directions

  • Velocity is tangent to the circle.
  • Centripetal acceleration points toward the center.
  • Centripetal means center-seeking.
  • For uniform circular motion, speed is constant but velocity changes direction.
Timing circular motion

2. Period and Frequency

Period and frequency describe how fast cycles repeat.

Period and frequency diagram Period measures time; frequency counts cycles one full revolution = T f = cycles per second
Period is the time for one cycle. Frequency tells how many cycles happen each second.
Period T = time for one cycle Unit: seconds.
Frequency f = cycles / time Unit: hertz, Hz.
Period-frequency relationship f = 1/T Also T = 1/f.

Examples

  • If a wheel makes 4 revolutions per second, frequency is 4 Hz.
  • If a satellite takes 5400 s for one orbit, its period is 5400 s.
  • If frequency increases, period decreases.
Angles and radians

3. Angular Motion

Angular quantities describe how an object rotates around a center. Radians are the natural angle unit for circular motion.

Angular motion diagram Radians connect angle to distance along an arc r theta arc length s s = r theta when theta is in radians
Use radians when connecting angle, radius, arc length, angular speed, and tangential speed.
Radians in a circle 1 revolution = 2π rad Also equals 360 degrees.
Arc length s = rθ θ must be in radians.
Angular speed ω = Δθ / Δt Unit: rad/s.
Angular speed and period ω = 2π / T For one complete revolution.
Angular speed and frequency ω = 2πf Frequency in hertz.
Linear-angular speed v = rω Connects tangent speed to angular speed.
Changing direction

4. Centripetal Acceleration

Centripetal acceleration is the inward acceleration needed to keep an object moving in a circular path.

Centripetal acceleration diagram Acceleration always points toward the center v v a_c points inward
The direction of velocity changes around the circle, so the acceleration direction is inward at every instant.
Centripetal acceleration ac = v2 / r Uses tangential speed and radius.
Angular form ac = rω2 Useful when angular speed is known.
Period form ac = 4π2r / T2 Useful when period and radius are known.
Common mistake: Centripetal acceleration is not caused by a special new force. It is the acceleration produced by the net inward force.
Net inward force

5. Centripetal Force

Centripetal force is not a separate kind of force. It is the name for the net inward force required for circular motion.

Centripetal force diagram A real inward force supplies the centripetal force v T or F_c inward Draw real forces first, then set net inward force = mv^2/r
On a string, tension points inward. On a road, friction may point inward. In orbit, gravity points inward.
Centripetal force Fc = mac Newton's second law for circular motion.
Speed-radius form Fc = mv2 / r Most common formula.
Angular form Fc = mrω2 Useful with angular speed.

Free-Body Diagram Rule

Do not draw "centripetal force" as an extra force unless the problem specifically labels a real force that way. Draw real forces first, then set the inward net force equal to mv2/r.

What provides center force?

6. Sources of Centripetal Force

Different situations use different real forces to provide the inward net force.

Sources of centripetal force diagram The inward force changes with the situation string tension road friction gravity
The equation may look like F = mv^2/r, but the actual inward force could be tension, friction, gravity, or a normal force.
Situation Real Force Toward Center Typical Equation
Ball on a stringTensionT = mv2/r
Car turning on flat roadStatic frictionfs = mv2/r
Satellite orbiting EarthGravityFg = mv2/r
Roller coaster loopWeight, normal force, or bothΣFinward = mv2/r
Object against circular wallNormal forceFN = mv2/r
Loops and changing force

7. Vertical Circles

In vertical circular motion, the force situation changes at the top, bottom, and sides of the circle because weight always points downward while inward direction changes.

Vertical circle force diagram At the top and bottom, inward changes direction mg and N can point inward N inward, mg downward center
Weight always points downward, but the inward direction depends on where the object is on the loop.
Top of loop mg + FN = mv2/r Both weight and normal force can point inward at the top.
Bottom of loop FN - mg = mv2/r Normal force inward/upward, weight downward.
Minimum top speed vmin = √(gr) For just maintaining contact at the top when FN = 0.
Common mistake: At the top of a loop, normal force can point downward because the seat or track pushes toward the center.
Turning without sliding

8. Banked Curves

A banked curve tilts the road so part of the normal force points toward the center of the turn. This helps vehicles turn without relying entirely on friction.

Banked curve diagram A tilted normal force has an inward component normal force inward component weight bank angle
Banking lets part of the normal force point inward, reducing how much friction is needed for the turn.
Ideal banked curve tan(θ) = v2 / rg For no friction needed.
Ideal speed v = √(rg tan(θ)) Speed that matches bank angle and radius.
Flat curve friction limit vmax = √(μsrg) For a flat curve where static friction supplies centripetal force.
Universal attraction

9. Universal Gravitation

Newton's law of universal gravitation says that every mass attracts every other mass. The force grows with mass and decreases with the square of distance.

Universal gravitation diagram Gravity is an attractive force between masses m1 m2 r double r -> force becomes one-fourth
Use center-to-center distance for r. Because gravity follows an inverse-square law, distance changes force strongly.
Universal gravitation Fg = Gm1m2 / r2 r is center-to-center distance.
Gravitational constant G = 6.67 × 10-11 N·m2/kg2 Universal constant.
Gravitational field strength g = GM / r2 Near Earth surface, g is about 9.8 m/s2.

Inverse-Square Meaning

If the distance between two masses doubles, gravitational force becomes one-fourth as strong. If the distance triples, force becomes one-ninth as strong.

Gravity as centripetal force

10. Orbital Motion

A satellite stays in orbit because gravity provides the centripetal force. The satellite is constantly falling toward Earth while moving forward fast enough to keep missing the ground.

Orbital motion diagram Gravity bends forward motion into an orbit v tangent gravity inward GmM/r^2 = mv^2/r
For a circular orbit, gravity is the inward force that produces centripetal acceleration.
Gravity provides circular force GmM/r2 = mv2/r Satellite mass cancels.
Orbital speed v = √(GM/r) For circular orbit around mass M.
Orbital period T = 2πr / v Time for one orbit.
Kepler-style relation T2 = 4π2r3 / GM For circular orbit.
Gravitational potential energy U = -GMm/r Often used in more advanced orbital energy problems.
Escape speed vesc = √(2GM/r) Minimum speed to escape without further propulsion.
Simulation labs

11. Simulation Labs for This Unit

These official PhET simulations help students visualize gravitational force, circular motion, orbit stability, orbital speed, mass effects, and multi-body interactions.

Gravity and Orbits

Explore how gravity controls circular and orbital motion for planets, moons, and satellites. Students can adjust mass, velocity, and distance to observe orbit changes.

Lab idea: increase orbital speed and observe whether the orbit becomes larger, smaller, or unstable.
Open Simulation
My Solar System

Build simple gravitational systems and observe how mass, initial speed, and position affect orbital paths and system stability.

Lab idea: create a stable two-body orbit, then change one body's velocity and record what happens.
Open Simulation
Investigation skills

12. Circular Motion and Gravitation Lab Skills

Labs in this unit often involve measuring radius, period, mass, speed, and force. Students should be able to compare measured centripetal force with the value predicted by mv2/r.

Common Labs

  • Whirling stopper or rubber stopper circular motion lab.
  • Turntable and friction lab.
  • Banked curve or flat curve model car lab.
  • Simulation-based orbit stability lab.
  • Period-radius relationship investigation.

Useful Measurements

  • Radius in meters.
  • Time for multiple revolutions.
  • Period and frequency.
  • Mass in kilograms.
  • Speed in meters per second.
  • Force in newtons.
Worked examples

13. Worked Examples

Example 1: Speed from period

A toy car moves in a circle of radius 2.0 m with period 4.0 s. Find tangential speed.

v = 2πr/T = 2π(2.0)/4.0 = 3.14 m/s.

Example 2: Centripetal acceleration

A 0.50 kg object moves at 6.0 m/s in a circle of radius 3.0 m. Find centripetal acceleration.

ac = v2/r = 6.02/3.0 = 12 m/s2.

Example 3: Centripetal force

Using the object in example 2, find centripetal force.

Fc = mac = (0.50)(12) = 6.0 N inward.

Example 4: Flat curve maximum speed

A car turns on a flat curve of radius 40 m. If μs = 0.60, find maximum speed before skidding.

Static friction provides centripetal force: μsmg = mv2/r.

vmax = √(μsrg) = √[(0.60)(40)(9.8)] = 15.3 m/s.

Example 5: Gravitational force

Two 1000 kg objects are separated by 2.0 m. Find the gravitational force between them.

F = Gm1m2/r2.

F = (6.67 × 10-11)(1000)(1000)/(2.0)2 = 1.67 × 10-5 N.

Example 6: Orbital speed

A satellite orbits Earth at radius 6.77 × 106 m from Earth's center. Use GMEarth = 3.99 × 1014 m3/s2. Find orbital speed.

v = √(GM/r).

v = √[(3.99 × 1014)/(6.77 × 106)] = 7.68 × 103 m/s.

Independent practice

14. Practice Problems

Try each problem first. Then open the answer check and compare formulas, units, and direction reasoning.

1. A wheel has radius 0.40 m and period 2.0 s. Find tangential speed.

Answer

v = 2πr/T = 2π(0.40)/2.0 = 1.26 m/s.

2. A fan spins at 5.0 revolutions per second. Find frequency and period.

Answer

f = 5.0 Hz. T = 1/f = 0.20 s.

3. An object moves at 10 m/s in a circle of radius 5.0 m. Find centripetal acceleration.

Answer

ac = v2/r = 100/5.0 = 20 m/s2.

4. A 2.0 kg object has centripetal acceleration 12 m/s2. Find centripetal force.

Answer

Fc = mac = (2.0)(12) = 24 N inward.

5. A 1.5 kg mass moves at 4.0 m/s on a string of radius 0.80 m. Find tension if tension is the only inward force.

Answer

T = mv2/r = (1.5)(4.0)2/0.80 = 30 N.

6. What direction is velocity in uniform circular motion?

Answer

Tangent to the circular path.

7. What direction is centripetal acceleration?

Answer

Toward the center of the circle.

8. A flat curve has radius 30 m and μs = 0.50. Find maximum speed.

Answer

vmax = √(μsrg) = √[(0.50)(30)(9.8)] = 12.1 m/s.

9. A banked curve has radius 50 m and angle 20 degrees. Find ideal speed with no friction.

Answer

v = √(rg tanθ) = √[(50)(9.8)tan(20)] = 13.4 m/s.

10. A roller coaster loop has radius 12 m. Find minimum speed at the top to maintain contact.

Answer

vmin = √(gr) = √[(9.8)(12)] = 10.8 m/s.

11. If orbital radius increases, what generally happens to orbital speed?

Answer

Orbital speed decreases for circular orbits around the same central mass.

12. Two masses attract with gravitational force F. If distance doubles, what is the new force?

Answer

F/4 because gravity follows an inverse-square relationship.

13. Two masses attract with force F. If one mass doubles and distance stays the same, what is the new force?

Answer

2F.

14. A satellite has orbital radius 8.0 × 106 m. Use GM = 3.99 × 1014 m3/s2. Find orbital speed.

Answer

v = √(GM/r) = √[(3.99 × 1014)/(8.0 × 106)] = 7.06 × 103 m/s.

15. What real force provides centripetal force for a satellite orbiting Earth?

Answer

Gravity.

16. In the Gravity and Orbits simulation, what happens if orbital speed is too low?

Answer

The object falls inward toward the central body instead of maintaining a stable orbit.

Final review

15. What to Know Before Moving On

  • Velocity is tangent to a circular path.
  • Centripetal acceleration points toward the center.
  • ac = v2/r and Fc = mv2/r.
  • Centripetal force is the net inward force, not a separate extra force.
  • Different real forces can provide centripetal force: tension, friction, gravity, normal force, or combinations.
  • For flat curves, static friction provides the inward force.
  • For satellites, gravity provides the inward force.
  • Universal gravitation follows F = Gm1m2/r2.
  • Orbital speed for a circular orbit is v = √(GM/r).
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